Variable Selection For Varying Coefficient Models With Heredity Constraint

This article mainly focuses on variable selection for Varying coefficient models with heredity constraint.Varying coefficient(VC)model is a generalization of ordinary linear model,which can not only retain the strong interpretability,but also has the flexibility of nonparametric model.When the dimensionality of the predictors is high,it is often supposed that the model is sparse,which means that some of the coefficient functions are nonzero.Besides the main effects,the interactions with heredity constraint are also consid-ered in this paper.We propose a unified variable selection method for VC models,which can simultaneously select the nonzero main effects and interaction effects,and estimate the unknown coefficient functions,meanwhile,the selected model enforces the hierarchical structure,that is,interaction terms can be selected into the model only if at least one of the corresponding main effects is in the model.Kernel method is employed to estimate the varying coefficient functions,and a combined overlapped Group Lasso regularization is introduced to implement variable selection to keep the hierarchical structure.Theorems proved that the proposed penalty estimators have the oracle properties in the theorems.Simulation studies and Boston housing data analysis demonstrate that the proposed method performs well on finite sample cases in terms of variable selection and estimation.